Maximum lilkelihood estimation in the $\beta$-model

We study maximum likelihood estimation for the statistical model for undirected random graphs, known as the $\beta$-model, in which the degree sequences are minimal sufficient statistics. We derive necessary and sufficient conditions, based on the polytope of degree sequences, for the existence of the maximum likelihood estimator (MLE) of the model parameters. We characterize in a combinatorial fashion sample points leading to a nonexistent MLE, and nonestimability of the probability parameters under a nonexistent MLE. We formulate conditions that guarantee that the MLE exists with probability tending to one as the number of nodes increases.

[1]  Albert-László Barabási,et al.  Statistical mechanics of complex networks , 2001, ArXiv.

[2]  Komei Fukuda,et al.  From the zonotope construction to the Minkowski addition of convex polytopes , 2004, J. Symb. Comput..

[3]  P. Holland,et al.  An Exponential Family of Probability Distributions for Directed Graphs , 1981 .

[4]  J. F. C. Kingman,et al.  Information and Exponential Families in Statistical Theory , 1980 .

[5]  Matthieu Latapy,et al.  Efficient and simple generation of random simple connected graphs with prescribed degree sequence , 2005, J. Complex Networks.

[6]  Allan Sly,et al.  Random graphs with a given degree sequence , 2010, 1005.1136.

[7]  Thomas M. Cover,et al.  Elements of Information Theory , 2005 .

[8]  M. M. Meyer.  Transforming Contingency Tables , 1982 .

[9]  Eric D. Kolaczyk,et al.  Statistical Analysis of Network Data: Methods and Models , 2009 .

[10]  Reuven Cohen,et al.  Complex Networks: Structure, Robustness and Function , 2010 .

[11]  Yi-Ching Yao,et al.  Asymptotics when the number of parameters tends to infinity in the Bradley-Terry model for paired comparisons , 1999 .

[12]  Stephen E. Fienberg,et al.  Algebraic Statistics for a Directed Random Graph Model with Reciprocation , 2009, 0909.0073.

[13]  Akimichi Takemura,et al.  MATHEMATICAL ENGINEERING TECHNICAL REPORTS Connecting Tables with Zero-One Entries by a Subset of a Markov Basis , 2009 .

[14]  Martina Morris,et al.  A Simple Model for Complex Networks with Arbitrary Degree Distribution and Clustering , 2006, SNA@ICML.

[15]  S. Fienberg,et al.  Categorical Data Analysis of Single Sociometric Relations , 1981 .

[16]  Steven M. Goodreau,et al.  Advances in exponential random graph (p*) models applied to a large social network , 2007, Soc. Networks.

[17]  Persi Diaconis,et al.  A Sequential Importance Sampling Algorithm for Generating Random Graphs with Prescribed Degrees , 2011, Internet Math..

[18]  Stephen E. Fienberg,et al.  An Exponential Family of Probability Distributions for Directed Graphs: Comment , 1981 .

[19]  A. Rinaldo,et al.  On the geometry of discrete exponential families with application to exponential random graph models , 2008, 0901.0026.

[20]  V. Sós,et al.  GRAPH LIMITS AND EXCHANGEABLE RANDOM GRAPHS , 2008 .

[21]  D. Geiger,et al.  On the toric algebra of graphical models , 2006, math/0608054.

[22]  T. Yan,et al.  HIGH DIMENSIONAL WILKS PHENOMENA IN RANDOM GRAPH MODELS , 2011 .

[23]  Richard P. Stanley,et al.  A Zonotope Associated with Graphical Degree Sequences , 1990, Applied Geometry And Discrete Mathematics.

[24]  Mark Newman,et al.  Networks: An Introduction , 2010 .

[25]  Mark E. J. Newman,et al.  The Structure and Function of Complex Networks , 2003, SIAM Rev..

[26]  Patrick J. Wolfe,et al.  Null models for network data , 2012, ArXiv.

[27]  Stephen E. Fienberg,et al.  Maximum Likelihood Estimation in Log-Linear Models: Theory and Algorithms , 2011 .

[28]  Alexander Schrijver,et al.  Theory of linear and integer programming , 1986, Wiley-Interscience series in discrete mathematics and optimization.

[29]  T. Yan,et al.  A central limit theorem in the β-model for undirected random graphs with a diverging number of vertices , 2012, 1202.3307.

[30]  Svante Janson,et al.  Threshold Graph Limits and Random Threshold Graphs , 2008, Internet Math..

[31]  Hisayuki Hara,et al.  Graver basis for an undirected graph and its application to testing the beta model of random graphs , 2011, 1102.2583.

[32]  M. M. Meyer,et al.  Statistical Analysis of Multiple Sociometric Relations. , 1985 .

[33]  Michael Joswig,et al.  polymake: a Framework for Analyzing Convex Polytopes , 2000 .

[34]  László Lovász,et al.  Limits of dense graph sequences , 2004, J. Comb. Theory B.

[35]  R. A. Bradley,et al.  RANK ANALYSIS OF INCOMPLETE BLOCK DESIGNS THE METHOD OF PAIRED COMPARISONS , 1952 .

[36]  Shelby J. Haberman,et al.  Maximum Likelihood Estimates in Exponential Response Models , 1977 .

[37]  Alexander I. Barvinok,et al.  The number of graphs and a random graph with a given degree sequence , 2010, Random Struct. Algorithms.

[38]  Garry Robins,et al.  An introduction to exponential random graph (p*) models for social networks , 2007, Soc. Networks.

[39]  R. A. Bradley,et al.  Rank Analysis of Incomplete Block Designs: I. The Method of Paired Comparisons , 1952 .

[40]  Shelby J. Haberman,et al.  An Exponential Family of Probability Distributions for Directed Graphs: Comment , 1981 .

[41]  Georg Rasch,et al.  Probabilistic Models for Some Intelligence and Attainment Tests , 1981, The SAGE Encyclopedia of Research Design.

[42]  S. Haberman,et al.  The analysis of frequency data , 1974 .

[43]  Eric D. Kolaczyk,et al.  Statistical Analysis of Network Data , 2009 .

[44]  O. Barndorff-Nielsen.  Information and Exponential Families in Statistical Theory , 1980 .

[45]  R. A. Bradley,et al.  RANK ANALYSIS OF INCOMPLETE BLOCK DESIGNS , 1952 .

[46]  E. Zermelo.  Die Berechnung der Turnier-Ergebnisse als ein Maximumproblem der Wahrscheinlichkeitsrechnung , 1929 .

[47]  H. Wynn,et al.  Algebraic Methods in Statistics and Probability II , 2001 .

[48]  Paul Erdös,et al.  On random graphs, I , 1959 .

[49]  M. Newman,et al.  Random graphs with arbitrary degree distributions and their applications. , 2000, Physical review. E, Statistical, nonlinear, and soft matter physics.

[50]  D. Hunter.  MM algorithms for generalized Bradley-Terry models , 2003 .

[51]  Peter L. Hammer,et al.  Difference graphs , 1990, Discret. Appl. Math..

[52]  M. Newman,et al.  Statistical mechanics of networks. , 2004, Physical review. E, Statistical, nonlinear, and soft matter physics.

[53]  Bernd Sturmfels,et al.  Commutative Algebra of Statistical Ranking , 2011, ArXiv.

[54]  N. Mahadev,et al.  Threshold graphs and related topics , 1995 .

[55]  Walter Willinger,et al.  Mathematics and the Internet: A Source of Enormous Confusion and Great Potential , 2009, The Best Writing on Mathematics 2010.

[56]  Lawrence D. Brown.  Fundamentals of Statistical Exponential Families , 1987 .

[57]  Erling B. Andersen,et al.  Discrete Statistical Models with Social Science Applications. , 1980 .

[58]  Stephen E. Fienberg,et al.  Maximum likelihood estimation in log-linear models , 2011, 1104.3618.

[59]  Christophe Weibel.  Implementation and Parallelization of a Reverse-Search Algorithm for Minkowski Sums , 2010, ALENEX.

[60]  H. A. David,et al.  The method of paired comparisons , 1966 .

[61]  L. R. Ford.  Solution of a Ranking Problem from Binary Comparisons , 1957 .

[62]  Jason Morton,et al.  Relations among conditional probabilities , 2008, J. Symb. Comput..

[63]  Gerhard H. Fischer,et al.  On the existence and uniqueness of maximum-likelihood estimates in the Rasch model , 1981 .

[64]  Edoardo M. Airoldi,et al.  A Survey of Statistical Network Models , 2009, Found. Trends Mach. Learn..

[65]  S. Fienberg,et al.  Log linear representation for paired and multiple comparisons models , 1976 .

[66]  Jacob G. Foster,et al.  Link and subgraph likelihoods in random undirected networks with fixed and partially fixed degree sequences. , 2006, Physical review. E, Statistical, nonlinear, and soft matter physics.

[67]  Mark E. J. Newman,et al.  Structure and Dynamics of Networks , 2009 .

[68]  Nicholas Eriksson,et al.  Polyhedral conditions for the nonexistence of the MLE for hierarchical log-linear models , 2006, J. Symb. Comput..

[69]  A. Agresti,et al.  Categorical Data Analysis , 1991, International Encyclopedia of Statistical Science.

[70]  Charles J. Geyer,et al.  Likelihood inference in exponential families and directions of recession , 2009, 0901.0455.

[71]  S. Lauritzen.  Exchangeable Rasch Matrices∗ , 2007 .